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This paper investigates dynamical processes for which the state of time t is described by a density function, and specifically dynamical processes for which the shape of the density becomes largely independent of the initial density as time increases. A sufficient condition (weak ergodic theorem) is given for this asymptotic similarity of densities. The processes investigated are in general time dependent, that is, nonhomogeneous in time. Our condition is applied to processes generated by expanding mappings on manifolds, piecewise convex transformations of the unit interval, and integro-differential equations.
Let A be a subset of R^{n} be a bounded open set with finitely many connected components A_{j} and let T be a smooth map on R^{n} with A a subset of T(A). Assume that for each A_{j }, A is a subset of T^{ k}(A_{j}) for all k sufficiently large. We assume that T is expansive, but we do not assume that T(A) = A. Hence for x in A, T ^{i} (x) may escape A as i increases. Let m_{ }be a smooth measure on A (with inf_{A}dm / dx 0) and let x in A be chosen at random (using m). Since T is expansive we may expect T ^{i}(x) to oscillate chaotically on A for a certain time and eventually escape A. For each measurable set E in A define m_{k} (A) to be the conditional probability that T^{k} (x) is in E given that x, T (x), ....T^{ k} (x) are in A. We show that m_{k} converges to a smooth measure m_{0} which is independent of the choice of m. One dimensional examples are stressed.
A class of piecewise continuous, piecewise C 1 transformations on the interval J with finitely many discontinuities n are shown to have at most n invariant measures.
A class of piecewise continuous, piecewise C 1 transformations on the interval [0,1] is shown to have absolutely continuous invariant measures.
A numerical procedure is described that can accurately compute the stable manifold of a saddle fixed point for a map of R2, even if the map has no inverse. (Conventional algorithms use the inverse map to compute an approximation of the unstable manifold of the fixed point.) We rigorously analyze the errors that arise in the computation and guarantee that they are small. We also argue that a simpler, nonrigorous algorithm nevertheless produces highly accurate representations of the stable manifold.
We discuss an algorithm to find the parameter value at which a nonlinear, dissipative, chaotic system undergoes crisis. The algorithm is based on the observation that at crisis, the unstable manifold of an unstable periodic point becomes tangent to the stable manifold of the same or another, related unstable periodic point. This geometric algorithm uses much less computation (or data) than estimating the critical parameter value by using the scaling relation for chaotic transients, (p-p)
In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is non-empty. The basin boundary is either smooth or fractal ( that is, it has a Cantor-like structure). When there are horseshoes in the basin boundary, the basin boundary is fractal. A relatively small subset of a fractal basin boundary is said to be accessible from a basin. However, these accessible points play an important role in the dynamics and, especially, in showing how the dynamics change as parameters are varied. The purpose of this paper is to present a numerical procedure that enables us to produce trajectories lying in this accessible set on the basin boundary, and we prove that this procedure is valid in certain hyperbolic systems.
A numerical procedure is described for computing the succesive images of a curve under a diffeomorphismof R^N. Give a tolerance , we show how to rigorously guarantee that each point on the computed curve lies no further than a distance from the "true" image curve. In particular, if is the distance between adjacent points (pixels) on a computer screen, then a plot of the computed curve coincides with true curve within the resolution of the display. A second procedure is described to minimize the amount of computation of parts of the curve that lie outside a region of interest. We apply the method to compute the one-dimensional stable and unstable manifolds of the Henon and Ikeda maps, as well as a Poincare map for the forced damped pendulum.
In dynamical systems examples are common in which there are regions containing chaotic sets that are not attractors, e.g. systems with horseshoes have such region. In such dynamical systems one will observe chaotic transients. An important problem is the Dynamical Restraint Problem: given a region that contains a chaotic set but contains no attractor find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time. We present two procedures (PIM triple procedures) for finding trajectories which stay extremely close to such chaotic sets for arbitrarily long periods of time.
Examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors. If almost every trajectory eventually leaves some regions, but the region contains a chaotic set, then typical trajectories will behave chaotically for a while and then will leave the region, and so we will observe chaotic transients. The main objective that will be addressed is the Dynamic Restraint Problem: Given a region that contains a chaotic set but does not contain an attractor, find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time. Systems with horseshoes have such regions as do systems with fractal basin boundaries, as does the Henon map for suitable chosen parameters. We present a numerical technique for finding trajectories which will stay in such chaotic sets for arbitrarily long periods of time and it leads to a "saddle straddle trajectory". The method is called the PIM triple procedure since it is based on so-called PIM triple. A PIM (Proper Interior Maximum) triple is three point (a, c, b) in a straight line segment such that the interior point c (i.e. c is between a and b) has the maximum escape time, that is, its escape time from the region is greater than the escape time of both a and b. Proper means the segment from a to b is smaller than a previously obtained segment. We show rigorously that the PIM triple procedure works in ideal situations. We find it works well even in less than ideal cases. This procedure can also be used for the computation of Lyapunov exponents. Furthermore, the accessible PIM triple procedure ( a refined PIM triple procedure for finding accessible trajectories on the chaotic saddle) will also be discussed.
A method is described for reducing noise levels in certain experimental time series. An attractor is reconstructed from the data using the time-delay embedding method. The method produces a new, slightly altered time series which is more consistent with the dynamics on the corresponding phase-space attractor. Numerical experiments with the two-dimensional Ikeda laser map and power spectra from weakly turbulent Couette-Taylor flow suggest that the method can reduce noise levels up to a factor of 10.
A method is proposed to transform a nonlinear differential system into a map without having to integrate the whole orbit as in the usual Poincare return map technique. It consists of constructing a piecewise linear map by coarse-graining the phase surface of section into simplices and using the Poincare return map values at the vertices to define a linear map on each simplex. The numerical results show that the simplicial map is a good approximation to the Poincare map and it leads to a factor of 20 to 40 savings in computer time as compared with the integration of the differential equation. Computation of the generalized information dimensions of a chaotic orbit for the simplicial map gives values in close agreement with those found for the Poincare map.
We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is constructive with probability one and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.
The effect of a long length scale static inhomogeneity on spiral wave dynamics is studied in the two-dimensional complex Ginzburg-Landau equation. We find that the inhomogeneity leads to the formation of a dominant spiral domain that suppresses other spiral domains, and that the spiral vortices slowly drift in the presence of an inhomogeneity with a velocity that is proportional to the local parameter gradients. We derive an expression for the spiral vortex drift velocity and present examples of both fixed point and limit cycle attractors of the spiral vortices.
The effect of weak inhomogeneity on spiral wave dynamics is studied within the framework of the two-dimensional complex Ginzburg-Landau equation description. The inhomogeneity gives spatial dependence to the frequency of spiral waves. This provides a mechanism for the formation of a dominant spiral domain that suppresses other spiral domains. The spiral vortices also slowly drift in the inhomogeneity, and results for the velocity are given.
The effect of adding a chiral symmetry breaking term to the two-dimensional Complex Ginzburg Landau equation is investigated. We find that this term causes a shift in the frequency of the spiral wave solutions and that the sign of this shift depends on the topological charge (handedness) of the spiral. For parameters such that nearly stationary spiral domains form (called a ``frozen'' state), we find that, due to this charge-dependent frequency shift, the boundary between oppositely charged spiral domains moves, resulting in the domination of one domains of charge over the other. In addition, we introduce a quantity which measures the chirality of patterns and use it to characterize the transition between frozen and turbulent states. We also find that, depending on parameters, this transition occurs in two qualitatively distinct ways.
In this paper, we investigate the stability of a straight vortex filament with phase twist described by the three-dimensional Complex Ginzburg-Landau Equation (CGLE). The results of the linear stability analysis show that the straight filament is stable in a limited region of the two parameter space of the CGLE. The stable region is dependent on the phase twist imposed on the filament and shrinks in size as the phase twist is increased. It is also shown numerically that the nonlinear evolution of an unstable initial straight filament can lead to a helical filament.
This paper examines the consequences of the interaction between temporal period doubling and spatial pattern formation. We propose a simple discrete time, spatially continuous system, where the discrete time dynamics incorporates period doubling and the spatial operator imposes patterning at a preferred length scale. We find that this model displays a variety of fiburcations between different spatio-temporal states and these bifurcations are generic in that they do not depend on the details of the model. The results from our simple model bear remarkable similarities with recent experiments on a vertically vibrated granular layer.
Cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field. In the case of the complex Ginzburg-Landau equation, disordered cells of effectively frozen spirals appear, separated by thin walls (shocks), on a scale much larger than the basic wavelength of the spirals. We show that these structures can be understood in very simple terms. In particular, we show that the walls are, to a good approximation, segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex. The fact that the phase is only defined up to an integer multiple of 2*pi introduces a quantization condition on the sizes of the smallest spiral domains. The transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations. The structure depends on the angle the wall makes with the local phase contours, and the behavior can be either monotonic or oscillatory, depending on the parameters.
Explicit asymptotic analytical results are derived for the motion of scroll wave filaments in the complex Ginzburg-Landau equation. Good agreement with numerical tests is obtained. The analysis highlights the necessity of allowing for previously ignored small wavenumber shifts in the propogation of the waves away from the filament.
An analytical treatment is presented for scroll waves in the complex Ginzburg-Landau equation in the limit of small filament curvature, torsion, and phase twist. Explicit expressions for the filament velocity and frequency shift are found. In addition, the analysis shows the existence of small wavenumber shifts in the propagation of the waves away from the filament; a feature not included in previous scroll wave theories. The theoretical results are verified numerically in the case of circular untwisted scroll rings and for straight and sinusoidal scroll filaments with phase twist.
A criterion for the reconnection of vortex filaments in the complex Ginzburg-Landau equation is presented. In particular, we give an estimate of the maximum intervortex separation beyond which coplanar filaments of locally opposite charge will not reconnect. This is done by balancing the motion of the filaments toward each other that would result if they were straight ( a two-dimensional effect ) with the opposing motion due to the filament curvature. Numerical experiments are in good agreement with the estimated vortex separation.
We consider the avalanche mixing of a monodisperse collection of granular solids in a slowly rotating drum. This process has been studied for the case where the drum rotates slowly enough that each avalanche ceases completely before a new one begins (METCALFE G., SHINBROT T., MCCARTHY J. J. and OTTINO J. M., Nature, 374 (1995)39). We develop a mathematical model for the mixing both in this discrete avalanche case and in the more useful case where the drum is rotated quickly enough to induce a continuous avalanche in the material but slowly enough to avoid significant inertial effects. When applied to the discrete case, our model yields results which are consistent with those obtained experimentally by Metcalfe et al.
Recent advances in the theory of the quasiclassical approximation for systems that are chaotic in the classical limit are extended to the case of ray splitting, in particular, to the splitting of an incident ray into a reflected and refracted component at a sharp interface. An instructive example is presented and novel results are found. These include evidence for ray splittng and periodic orbits in the spectral correlations and a new type of "scarred" eigenstate based on combining nonisolated periodic orbits whose quasiclassical contributions have a nontrivial phase from total internal reflection.
Ray splitting is a phenomenon whereby a ray incident on a boundary splits into more than one ray traveling away from the boundary. The most common example of this is the situation, originally considered by Snell in 1621, in which an incident l ight ray splits into reflected and transmitted rays at a discontinuity in refractive index. This paper seeks to extend techniques and results from quantum chaos to short wavelength problems in which ray splitting surfaces are present. These extensions are tested using a simple model problem for the Schrödinger equation in two dimensions with a finite step potential discontinuity. Numerical solutions for the energy spectrum and eigenfunctions in this system are then compared with predictions based on quasiclassical theoretical results suitably extended to include ray splitting. Among the topics treated are the ray orbits for our problem, energy level statistics, scars, trace formulas, the quasiclassical transfer operator technique, and the effect of l ateral waves. It is found that these extensions of quantum chaos are very effective for treating problems with ray splitting.
Ray splitting is the phenomenon whereby a ray incident on a boundary splits into more than one ray traveling away from the boundary. Motivated by the recent application of ideas of quantum chaos to cases with ray splitting, we present an analy sis of the smoothed density of states for two-dimensional billiardlike systems with ray splitting. Using a simple heuristic technique, we obtain a contribution (analogous to the usual perimeter contribution) that is proportional to the length of the ray s plitting boundary. The result is expressed in a general form, allowing application to a variety of physical situations. A comparison is also made of the analytical result with numerical data from a particular example.
The properties of "scars" on eigenfunctions (i.e. enhancements along unstable classical periodic orbits) of a two-dimensional, classically choatic billiard are studied. It is shown that the tendency for a scar to form is controlled by both the stability of the periodic orbit and the statistical fluctuations in the time for wave density to return to the unstable orbit once having left. Both scars and "antiscars" are predicted to cocur depending on the nearness of the eigenvalue of the chaotic e igenfunction in question to a value that quantizes the periodic orbit. The theoretical predictions are compared with direct numerical solutions for a bowtie shaped billiard.
A family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with intermingled basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.
When a chaotic attractor lies in an invariant subspace, as in systems with symmetry, riddling can occur. Riddling refers to the situation where the basin of a chaotic attractor is riddled with holes that belong to the basin of another attracto r. We establish properties of the riddling bifurcation that occurs when an unstable periodic orbit embedded in the chaotic attractor, usually of low period, becomes transversely unstable. An immediate physical consequence of the riddling bifurcation is th at an extraordinarily low fraction of the trajectories in the invariant subspace diverge when there is a symmetry breaking.
Recent studies have revealed that riddled fractal sets, sets whose conventionally defined fractal dimensions are integers, occur commonly in chaotic dynamical systems. We demonstrate that these exotic fractal sets exhibit a sign-singular scali ng behavior with nontrivial scaling exponents. The exponents may then be used to characterize the sets. Numerical examples using both a low-dimensional map and a coupled map lattice are given.
We consider situations where a nonlinear dynamical system possesses a smooth invariant manifold. For parameter values p less than a critical value p, the invariant manifold has within it a chaotic attractor of the system. As p increases through p a blowout bifurcation takes place, in which the former attraction to the manifold changes to repulsion, and the chaotic set in the manifold ceases to be an attractor of the system. Depending on the dynamics away from the manifold, blowout bifurcations can be either hysteretic or nonhysteretic, and they are correspondingly accompanied wither by riddled basins (in which the basin is a "fat fractal") or by an extreme form of temporally intermittent bursting recently called on-off intermittency. The role of the dynamics away from the manifold in determining the hysteretic or supercritical nature of the bifurcation is explicitly illustrated with a numerical example.
Recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another basin arbitrarily nearby (the basin is "riddled" with holes). Here we consider the dynamics near the transition to this situation as a parameter is varied. Using a simple analyzable model, we obtain the characteristic behaviors near this transition. Numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered.
Recently it has been shown that there are chaotic attractors whose basins are such that every point in the attractor's basin has pieces of another attractor's basin arbitrarily nearby (the basin is "riddled" with holes). Here we repor t quantitative theoretical results for such basins and compare with numerical experiments on a simple physical model.
Theory and examples of attractors with basins which are of positive measure, but contain no open sets, are developed; such basins are called riddled. A theorem is established which states that riddled basins are detected by normal Lyapunov exponents. Several examples, both mathematically rigorous and numerical, motivated by applications in the literature, are presented.
The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In particular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the unstable (and stable) tangent spaces are not constant over the chaotic set; we call this unstable dimension variability. A simple two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed.
We discuss a topological property which we believe provides a useful conceptual characterization of a variety of strange sets occurring in nonlinear dynamics (e.g., strange attractors, fractal basin boundaries, and stable and unstable manifolds of chaotic saddles). Sets with this topological property are known as indecomposable continua. As an example, we give detailed results for the case of an indecomposable continuum that arises from the entrainment of dye advected by a fluid flowing past a cylinder. We show for this case that the indecomposable continuum persists in the presence of small noise.
We study the existence or nonexistence of the true trajectories of chaotic dynamical systems that lie close to computer-generated trajectories. The nonexistence of such shadowing trajectories is caused by finite-time Lyapunov exponents of the system fluctuating about zero. A dynamical mechanism of the unshadowability is explained through a theoretical model and identified in simulations of a typical physical system. The problem of fluctuating Lyapunov exponents is expected to be common in simu lations of higher-dimensional systems.
We present a technique for constructing a computer-assisted proof of the reliability of a long computer-generated trajectory of a dynamical system. Auxiliary calculations made along the noise-corrupted computer trajectory determine whether the re exists a true trajectory which follows the computed trajectory closely for long times. A major application is to verify trajectories of chaotic differential equations and discrete systems. We apply the main results to computer simulations of the Henon map and the forced damped pendulum.
For a chaotic system, a noisy trajectory diverges rapidly from the true trajectory with the same initial condition. To understand in what sense the noisy trajectory reflects the true dynamics of the actual system, we developed a rigorous proce dure to show that some true trajectories remain close tot he noisy one for long times. The procedure involves a combination of containment, which establishes the existence of an uncountable number of true trajectories close to the noisy one, and refinemen t, which produces a less noisy trajectory. Our procedure is applied to noisy chaotic trajectories of the standard map and the driven pendulum.
This paper deals with the problem : Can a noisy orbit be tracked by a real orbit? In particular, we will study the one-parameter family of tent maps and the one-parameter family of quadratic maps. We write gm for either fm or Fm with fm (x) = mx for x <= 1/2 and fm (x) = m(1-x) for x =1/22, and Fm (x) = mx (1-x). For a given m we will say: gm permits increased parameter shadowing if for each deltax 0 there exists some deltam 0 and some deltaf 0 such that every deltaf -pseudo gm -orbit starting in some invariant interval can be deltax -shadowed by a real ga -orbit with a = m + deltam. We show that gm typically permits increased parameter shadowing.
We study the family of tent maps - continuous, unimodal, piecewise linear maps of the interval with slopes absolute value s, sqrt (2) <= s <=2. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters s, although they fail to have the shadowing property for an uncountable, dense set of parameters. We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.
Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. In an iterated process, if errors double each iterate and numerical calculations have 50-bit (or 15-digit) accuracy, a true orbit through a point can be expected to have no correlation with a numerical orbit after 50 iterates. On the other hand, numerical studies often involve hundreds of thousands of iterates. One may therefore question the validity of such studies. A relevant result in this regard is that of Anosov and Bowen who showed that systems which are uniformly hyperbolic will have the shadowing property: a numerical (or noisy) orbit will stay close to (shadow) a true orbit for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report rigorous results for nonhyperbolic systems: numerical orbits typically can be shadowed by true orbits for long time periods.
We consider simply Lyapunov-exponent based conditions under which the response of a systems to a chaotic drive is a smooth function of the drive state. We call this differentiable generalize synchronization (DGS). When DGS does not hold, we quantify the degree of nondifferentiability using the Holder exponent. We also discuss the consequences of DGS and give an illustrative numerical example.
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